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From: kerber@uklirb.UUCP (Manfred Kerber)
Newsgroups: comp.ai
Subject: Re: Fun with the semantics of paradox
Keywords: Types, Paradoxes
Message-ID: <3781@uklirb.UUCP>
Date: 30 Jan 89 09:43:15 GMT
References: <1883@buengc.BU.EDU> <2996@uhccux.uhcc.hawaii.edu> <905@ubu.warwick.UUCP> <479@aipna.ed.ac.uk> <1036@hudson.acc.virginia.edu> <3715@uklirb.UUCP> <48717@yale-celray.yale.UUCP>
Reply-To: kerber@uklirb.UUCP (Manfred Kerber)
Organization: Universitaet Kaiserslautern, West Germany
Lines: 40

In answer to Blair P. Houghton:
>>>>    
>>>>       "The following sentence is true"
>>>>       "The preceeding sentence is false"    ?
>>
>How bout a few symbols?
>
>1.  S1 ==> S2
>           __
>2.  S2 ==> S1

No, this notion is not sufficient, because one has also 1.' NOT S1 ==> NOT S2,
analogous for 2. So in both cases one has a ``<==>'' instead of a ``==>''.
One gets S1 <==> NOT S1. It is a real paradox.


In answer to Sean Philip Engelson:
>In article <3715@uklirb.UUCP>, Manfred Kerber resolves the paradox by
>introducing Russell's hierarchy of types, saying that the first is of
>types "Sentence about Sentence", thus the second must thus be of type
>"Sentence", but it's also "S about S".  However, if you allow infinite
>types, the paradox remains, as both sentences can be of type T, where
>
>T is defined as the fixed point of T', as follows:
>      T' = "Sentence" | "Sentence about T'"
>Each sentence is of type T and can thus refer to the other.  Is there
>any a priori reason to exclude infinite types?
No, there is no a priori reason to exclude infinite types, but the
paradoxies of Russell can be a reason to avoid self-refential assertions.
One method is to use a strict hierarchy of types. This does not exclude
to use infinitely many types, but your fix point construction is not okay.
I see no problem to use types associated to each ``ordinal number'' 
inclusive omega ordinals, as e.g. by:
T(1) = "Sentence", T(i) = "Sentence about Sentence of Type T(k) with k<i"
So one can have types:
T(1),T(2),T(3),...,T(omega),T(omega+1),T(omega+2),... and so on, where
omega is the ordinal associated to N, the set of natural number.
Indeed one gets a weaker system as Andrew C.Plotkin remarks. An assertion
as "This sentence is true" is excluded without being paradox.
Manfred Kerber